/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 If \(\widehat{\beta}_{0}\) and \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 1

If \(\widehat{\beta}_{0}\) and \(\widehat{\beta}_{1}\) are the least-squares estimates for the intercept and slope in a simple linear regression model, show that the least-squares equation \(\hat{y}=\hat{\beta}_{0}+\hat{\beta}_{1} x\) always goes through the point \((\bar{x}, \bar{y})\). [Hint: Substitute \(\bar{x}\) for \(x\) in the least-squares equation and use the fact that \(\left.\widehat{\beta}_{0}=\bar{y}-\widehat{\beta}_{1} \bar{x} .\right]\)

Short Answer

Expert verified
The least squares line always passes through the mean point \((\bar{x}, \bar{y})\).

Step by step solution

01

Substitute into the Least-Squares Equation

Begin with the equation for the least squares line: \[\hat{y} = \widehat{\beta}_{0} + \widehat{\beta}_{1} x\]Substitute \(\bar{x}\) for \(x\):\[\hat{y} = \widehat{\beta}_{0} + \widehat{\beta}_{1} \bar{x}\]
02

Express the Intercept using Means

According to the hint, use the relationship for the intercept:\[\widehat{\beta}_{0} = \bar{y} - \widehat{\beta}_{1} \bar{x}\]Substitute this expression into the equation for \(\hat{y}\):\[\hat{y} = (\bar{y} - \widehat{\beta}_{1} \bar{x}) + \widehat{\beta}_{1} \bar{x}\]
03

Simplify the Expression

Simplify the right side of the equation:\[\hat{y} = \bar{y} - \widehat{\beta}_{1} \bar{x} + \widehat{\beta}_{1} \bar{x}\]The terms \(- \widehat{\beta}_{1} \bar{x}\) and \(+ \widehat{\beta}_{1} \bar{x}\) cancel each other out, leaving:\[\hat{y} = \bar{y}\]
04

Conclusion

We have shown that substituting \(\bar{x}\) into the least squares equation results in \(\hat{y} = \bar{y}\). Thus, the least squares line passes through the point \((\bar{x}, \bar{y})\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Least-Squares Estimates
In simple linear regression, least-squares estimates are crucial to finding the best fitting line through a set of data points. This method minimizes the sum of the squared differences between the observed values and the values predicted by the line. Here's how it works:
  • The line is defined by the equation \( \hat{y} = \widehat{\beta}_{0} + \widehat{\beta}_{1} x \), where \( \widehat{\beta}_{0} \) is the intercept and \( \widehat{\beta}_{1} \) is the slope.
  • The goal is to find the values of \( \widehat{\beta}_{0} \) and \( \widehat{\beta}_{1} \) that minimize the sum of square differences: \( \sum (y_i - (\widehat{\beta}_{0} + \widehat{\beta}_{1} x_i))^2 \).
  • To achieve this, calculus is typically used to find the minimum point of this function, giving the least-squares estimates for the intercept and slope.
This method ensures that the line is as close as possible to all data points, providing a precise fit for the given data.
The Role of Intercept and Slope
The intercept and slope are two foundational elements of the simple linear regression equation, \( \hat{y} = \widehat{\beta}_{0} + \widehat{\beta}_{1} x \). Understanding their roles is key to grasping how linear regression works.
  • The intercept, \( \widehat{\beta}_{0} \), represents the predicted value of \( y \) when \( x = 0 \). It's a starting point on the \( y \)-axis for the line being drawn.
  • The slope, \( \widehat{\beta}_{1} \), indicates how much \( y \) is expected to increase when \( x \) increases by one unit. This tells us about the relationship's direction and strength.
Together, these components describe how changes in \( x \) are expected to impact \( y \), shaping the line that most accurately represents the data pattern.
Significance of the Mean Point
In the context of least-squares estimates, the mean point \( (\bar{x}, \bar{y}) \) holds special significance. It plays a central role in validating the fit of the regression line. Here's how:
  • The mean point represents the averages of all \( x \) values and all \( y \) values, providing a central location in the data set.
  • The least-squares regression line always passes through this mean point. This is a useful check to ensure that the regression calculation is done correctly.
  • To see why the line passes through \( (\bar{x}, \bar{y}) \), consider substituting \( \bar{x} \) into the least-squares equation. As shown, doing so results in \( \hat{y} = \bar{y} \), proving that the line indeed crosses the mean point.
This ensures that the regression model effectively balances above and below the data, making it a reliable representation of the overall trend.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Utility companies, which must plan the operation and expansion of electricity generation, are vitally interested in predicting customer demand over both short and long periods of time. A short-term study was conducted to investigate the effect of each month's mean daily temperature \(x_{1}\) and of cost per kilowatt-hour, \(x_{2}\) on the mean daily consumption (in \(\mathrm{kWh}\) ) per household. The company officials expected the demand for electricity to rise in cold weather (due to heating), fall when the weather was moderate, and rise again when the temperature rose and there was a need for air conditioning. They expected demand to decrease as the cost per kilowatt-hour increased, reflecting greater attention to conservation. Data were available for 2 years, a period during which the cost per kilowatt-hour \(x_{2}\) increased due to the increasing costs of fuel. The company officials fitted the model $$Y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{1}^{2}+\beta_{3} x_{2}+\beta_{4} x_{1} x_{2}+\beta_{5} x_{1}^{2} x_{2}+\varepsilon$$ to the data in the following table and obtained \(\hat{y}=325.606-11.383 x_{1}+.113 x_{1}^{2}-21.699 x_{2}+.873 x_{1} x_{2}-.009 x_{1}^{2} x_{2}\) with \(\mathrm{SSE}=152.177\) When the model \(Y=\beta_{0}-\beta_{1} x_{1}+\beta_{2} x_{1}^{2}+\varepsilon\) was fit, the prediction equation was \(\hat{y}=130.009-3.302 x_{1}+.033 x_{1}^{2}\) with \(\mathrm{SSE}=465.134 .\) Test whether the terms involving \(x_{2}\left(x_{2}, x_{1} x_{2}, x_{1}^{2} x_{2}\right)\) contribute to a significantly better fit of the model to the data. Give bounds for the attained significance level.

The correlation coefficient for the heights and weights of ten offensive backfield football players was determined to be \(r=.8261\) a. What percentage of the variation in weights was explained by the heights of the players? b. What percentage of the variation in heights was explained by the weights of the players? c. Is there sufficient evidence at the \(\alpha=.01\) level to claim that heights and weights are positively correlated? d. What is the attained significance level associated with the test performed in part (c)?

Television advertising would ideally be aimed at exactly the audience that observes the ads. A study was conducted to determine the amount of time that individuals spend watching TV during evening prime-time hours. Twenty individuals were observed for a 1 -week period, and the average time spent watching TV per evening, \(Y\), was recorded for each. Four other bits of information were also recorded for each individual: \(x_{1}=\) age, \(x_{2}=\) education level, \(x_{3}=\) disposable income, and \(x_{4}=\) IQ. Consider the three models given below: Model I: $$Y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}+\beta_{4} x_{4}+\varepsilon$$ Model II: $$Y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\varepsilon$$ Model III: $$Y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{1} x_{2}+\varepsilon$$ Are the following statements true or false? a. If Model I is fit, the estimate for \(\sigma^{2}\) is based on 16 df. b. If Model II is fit, we can perform a \(t\) test to determine whether \(x_{2}\) contributes to a better fit of the model to the data. c. If Models I and II are both fit, then \(\mathrm{SSE}_{\mathrm{I}} \leq \mathrm{SSE}_{\mathrm{II}}\) d. If Models I and II are fit, then \(\hat{\sigma}_{1}^{2} \leq \widehat{\sigma}_{\Pi}^{2}\). e. Model II is a reduction of model I. f. Models I and III can be compared using the complete/reduced model technique presented in Section 11.14.

Consider the general linear model $$Y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\cdots+\beta_{k} x_{k}+\varepsilon$$ where \(E(\varepsilon)=0\) and \(V(\varepsilon)=\sigma^{2} .\) Notice that \(\widehat{\beta}_{i}=\mathbf{a}^{\prime} \hat{\boldsymbol{\beta}},\) where the vector a is defined by $$a_{j}=\left\\{\begin{array}{ll} 1, & \text { if } j=i, \\ 0, & \text { if } \neq i \end{array}\right.$$ Use this to verify that \(E\left(\widehat{\beta}_{i}\right)=\beta_{i}\) and \(V\left(\widehat{\beta}_{i}\right)=c_{i i} \sigma^{2},\) where \(c_{i i}\) is the element in row \(i\) and column \(i\) of \(\left(\mathbf{x}^{\prime} \mathbf{x}\right)^{-1}\)

The data in the following table give the miles per gallon obtained by a test automobile when using gasolines of varying octane levels.$$\begin{array}{cc} \hline \text { Miles per Gallon }(y) & \text { Octane }(x) \\ \hline 13.0 & 89 \\\13.2 & 93 \\\13.0 & 87 \\\13.6 & 90 \\\13.3 & 89 \\\13.8 & 95 \\\14.1 & 100 \\\14.0 & 98\\\\\hline\end{array}$$a. Calculate the value of \(r\). b. Do the data provide sufficient evidence to indicate that octane level and miles per gallon are dependent? Give the attained significance level, and indicate your conclusion if you wish to implement an \(\alpha=.05\) level test.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.